IL NUOVO CIMENT0 VOL. 23B, N. 2 l l 0 t tobre 1974

Infra-Red Polarized Emission from Anisotropic Cosmic

S. AIELLO~ A. ~BO]NETTI a n d F . ME~CARAGLIA

Cattedra di .Fisica dello Spaz io e Uni t~ di Ricerca C N R . G I F C O - .Firenze I s t i tu to di P i s i ca dell 'Universi t~t - .Firenze

(ricevuto il 12 Marzo 1974)

%

Dust.

Summary. - - The problem of polarimetric effects of cosmic dust has been reviewed, with particular reference to the magnetic alignment mechanism. Aligned anisotropic (lust grains are expected to emit detectable polarized radiation in the far infra-red region. Polarimetric infra-red measurements can contribute effectively to the knowledge of the nature and distribution of dust grains.

1 . - I n t r o d u c t i o n .

1"1. - Interstel lar and circumstellar dust was firstly in t roduced to explain the selective absorption of visible and ultraviolet starlight. More recently,

direct evidence of dust existence was obtained observing the ilffra-red fluxes from several sources, bo th galactic and extragalactic. I n most computa t ions

made up to now to explain such fluxes, dust grains have been assumed to be isotropic both in shape and optical properties (1).

This is however inconsistent with the observed polarizing properties of

dust clouds. A great deal of observational work has been done in visible

(1) See , for instance, a) W. S~]~I~: Astrophys . Journ . , 144, 318 (1966); b) W. STEI~: Astrophys . Journ . , 145, 101 (1966) ; c) S. AI]~LLO and A. BORGHESI: Quadern i della _Ricerca Scient i] ica del C2VR, 64, 29 (1970); d) K. S. KRISH~A SWAM]:: Astrophys . Journ . , 167, 63 (1971); e) M. R. BIASI: Thesis, Univcrsity of Leccc (1972).

20 - II Nuovo Cimento B. 297

298 S. AIELLO, A. BONETTI and F. MENCARAGLIA

astronomical polar imetry (2). The origin of starlight polarization m a y be both

interstellar (dust distributed more or less uniformly between star and observer) and circumstellar (dust distr ibuted in a more or less restr icted region a round the star). I n some cases polarization appears to be t ime dependent (3). Much

less work has been done up to now in infra-red polar imetry (lb.4).

1"2. - The aim of this paper is to discuss in some detail the possibility of observing infra-red polarized fluxes. The present t r ea tmen t deals in par t icular with the problem of general interstellar polarization.

Section 2 of this paper will be devoted to a brief discussion of the relation- ship between starlight polarization and large-scale dust distr ibution in the galactic structure.

I n Sect. 3 we apply the magnet ic mechanism of al ignment to a suitable model of dust and compare the results with the observational data. Section 4

presents preliminary computat ions of the far infra-red polarized fluxes expected from dust uggregations. Some final remarks and conclusions are found in Sect. 5.

2. - Starlight polarizat ion and dust distribution in galact ic structure.

2"1. - Starlight polarization is a t t r ibu ted to dust grains, assuming t h a t

grains are anisotropic in shape and/or in optical properties and t h a t t h e y are aligned by suitable mechanisms.

Detailed maps of the distribution of intensi ty and direction of the polarized component of starlight on a galactic scale have been published (2a.b.o.~). Some

striking features are evident. First of all, the polarization directions are not

everywhere randomly distributed, bu t th roughou t large sections of galactic

(2) a) W. A. HILTNER: Science, 109, 166 (1949); Astrophys. Journ. Suppl. II , 24, 389 (1956); b) J. S. HALL: Pubbl. US Nay. Obs., 17, 1 (1950); c) G. KLARE and T. NECK:EL: I A U Symposium No. 38, 449 (1970); Astron. Astrophys., l l , 155 (1971); d) D. A. MATHEWSON and V. L. FORD: Mere. Roy. Astr. Soc., 75, 139 (1970); e) K. S]~RKOWSKY and J. W. ROBERTSON: Astrophys. Journ., 158, 441 (1969); /) K. SER- •OWSKI: Astrophys. Journ., 160, 1081 (1970); g) P. G. MARTIN, R. ILLING and R. G. P. ANG]~L: .Mont. Not. Roy. Astr. Soc., 159, 191 (1972). i) For a. review, see J. M. GREENBERG: .57ebulae and Interstellar .Matter, edited by B. M. MIDDLEHURST and L. H. ALLER (Chi- cago, 1968); and .Proceedings o/ the I Symposium on Astronomical Photopolarimetry (Tucson, Ariz., 1972). (a) I. APPENZELLER and W. A. HILTNER: Astrophys. Journ., 149, 353 (1967); B. ZELLN~R: Astrophys. Journ., 75, 182 (1970). (4) R.W. CAPPS and H. M. DICK: Astrophys. Journ., 175, 693 (1972); P. G. MARTIN: Astrophys..Eett., 7, 193 (1972); F. J. Low, D. E. KLEINMANN, F. F. FOXB]~S and H. H. AUMANN: Astrophys. Journ., 157, L97 (1969); G. DALL'OGLIO, B. MELCHIORRI, F. MELCH1ORRI, V. NATALE, S. AIELLO and F. MENCARAGLI•: Far In/raved Polarimetry (Tucson, Ariz., 1972) (see ref. (2i)).

INFRA-RED POLARIZED ~MISSION FROM ANISOTI:tOPIC COSMIC DUST 2 ~

longitude t hey follow a regular p a t t e r n , being a lmost parallel a m o n g them- selves, and to the galactic plane.

This supports the view t h a t an a l ignment mechan i sm exists and is regular over large regions, being re la ted to the galactic s t ructure . A plausible mecha- nism indicated long ago (5) is a general magne t ic field, essential ly aligned wi th the a rm axes. This m a y not be efficient enough to ensure the required degree of a l ignment ; other mechanisms scan be associated with the a r m s t ruc ture , such as photon in terac t ion (~), or s t r eaming gas (~), or cosmic-ray in te rac t ion (s).

Less clear is the relat ionship be tween polar iza t ion and the a m o u n t of inter- stellar m a t t e r along the line of sight, except t h a t polar izat ion is smaller where the amount of m a t t e r is small. The opposi te is not t rue, bo th for i n t ens i ty and for regular i ty of direction. Also not s imple is the re la t ionship be tween polar izat ion and ext inct ion (see f.i. (~)).

2"2. - As s ta ted by Boca;, (( clumpiness is a basic character is t ic of in ters te l lar m e d i u m ~) (3). In te rs te l la r dust is ma in ly found in concent ra t ions such as clouds and lanes of dark nebulosities. Dus t lanes seem to follow the spiral p a t t e r n of the ga laxy; indeed (( dust lanes of ga laxy be t t e r define spiral pa t t e rns t h a n do H I I regions ~) (s0); these appea r to be located within and a t the edge of dus t lanes.

I n the solar ne ighbourhood the dust lane could be ( 3 0 0 - - 5 0 0 ) p c s wide towards the inner edge of the a r m (1~): these dimensions should be c o m p a r e d wi th those of dark nebulae (dense; several magn i tudes local ext inct ion), which range f rom 0.01 pc (Bock globules) to several pcs (such as the da rk nebula to the south of Q-Oph (10~)), and tens of pcs, as the recent ly observed g ian t dust complex in the Perseus a rm (~2).

A convenient large-scale model of dust d is t r ibut ion m a y consist of a (~ quasi-homogeneous ~, low-densi ty componen t ( the opaque p a r t of the general inters tel lar m a t t e r ; ext inct ion comparab le to the average ex t inc t ion per uni t distance) along the spiral arms, wi th super imposed the small-scale, large-densi ty , s t ructures of the various kinds of da rk nebulae (*). I t seems reasonable to re la te the large-scale regularit ies of s tar l ight polar iza t ion to the quas i -homogeneous componen t of dust (and magne t ic field) along the spiral arms, while local

(5) a) L. Sl~ITZ]~R and J. TuI(•Y: Science, 109, 461 (1949); b) L. DAws and J. L. GREENSTEIN: Astrophys. Journ., 114, 206 (1951). (6) M. HARW~T: ~Vature, 226, 61 (1970). (7) T. GOLD: Mont. Not. Roy. Astr. Soc., 112, 215 (1952). (8) E. E. SALFET]~R and N. C. WICKR.~MASINGn]~: Nature, 222, 242 (1969). (9) B . J . BOCK and G. S. CORDWELL: A study o] dar]~ nebulae, Report of the Steward Observatory, University of Arizona (1971). (10) B. T. LYNDS: I A U Symposium No. 38, 26 (1970). (11) j . M. GRE]~NB~.RG: I A U Symposium No. 39, 306 (1970). (13) B. H(SGLUND and M. A. GORDON: Astrophys. Journ., 182, 45 (1973). (*) These dense clouds (( are qualitatively different from the HI clouds )) to which the present work refers (see (15~), p. 401).

3 0 0 S. AIELLO, A. BONETTI a n d F. ME:NCARAGLIA

effects are expected to take place in dust aggregations of more than average size and density.

2"3. - While there is a good deal of information on the denser clouds, much

scarcer information is available for the dilute-dust component and this because of the difficulty of its observation. There are suggestions f rom GI~EENBERG (11) which lead to the conclusion tha t grain growth is unlikely to take place in interarm regions, where there are hot ( (1000- -10000)~ H I regions, while particles with radius a ~ 1 0 -5 cm may grow in normal cold H I regions. The t ime scale of growth is given as 3.107 years, which is about the t ime spent

by the arm gas in the dust lane. That gas and dust are generally mixed has been proven b y observations

beyond doubt. Whatever are the mechanisms of dis tr ibut ion and mixing of

gas and dust, it seems to us tha t a different grain composition m a y be expected in different regions of the spiral arms, as a consequence of local dust product ion

from such mechanisms as late-type stars, and supernova and nova explosions. The observations of WnITTET et al. (~3) seem to confirm the existence of such

local differences. On the other hand, the low-density dust component is p resumably affected

by a considerable degree of inhomogenei ty : experimental evidence shows tha t the observed polarization could be explained in terms of small knots of dust, whose dimensions ( ~ 1 pc) and number ( ~ 4 0 per pc along the line of sight)

are of the same order of magni tude as deduced from interstellar reddening (~4). The dust clumpiness parallels the inhomogenei ty of the magnet ic field (~),

and is most likely related to the elumpiness associated with the two-phase structure of interstellar gas in H I regions (<, raisin pudding ~> model (~5)).

The interaction of dust with the hot and cold components of interstellar

gas is dealt with in the following Section, with reference to the problem of

grain alignment.

3. - Degree of al ignment and polarization efficiency.

3"1. - Several models for interstellar light polarization have been suggested. Following WICKRAMASINGHE (16) the main source of interstellar polarizat ion

Qs) D. C. B. WHITTET, I. C. VAN BREDA and K. NANDY: Nature Phys. Sci., 243, 21 (1973). (14) K. SERKOWSK~: Astrophys. Journ., 154, 115 (1968). (15) a) B. G. CLARKE: Astrophys. Journ., 142, 1398 (1971); b) V. RADAKRISI-INAN,

J. D. MURRAY, P. LOCKHART and R. P. J. WHIT~L~: Astrophys. Journ., Suppl. No. 203, papers II and V (1972); M. P. HUGHES, A. R. THOMSON and R. S. COLVlN: Astrophys. Journ. Suppl., 23, 323 (1971); Review articles: c) G. B. FIELD: in Molecules in the Galactic Environment (New York, N.Y., 1973), p. 22; d) A. DALGARNO and R. A. McCRAY: Ann. t~ev. Astr. Astrophys., 10, 375 (1972). (16) N, C. WICKRAMASINGHE: Nature, 224, 656 (1969).

I N ~ R A - R E D POLARIZED EMISSION FROM A N I S O T R O P I C COSMIC D U S T 3 ~ 1

are silicate grains wi th a small cont r ibut ion f rom graphi te grains, as this m ix tu r e explains f,~irly well the observed wave leng th dependence of inters te l lar polar- izat ion in the visible region. I n this model silicate gr~ins ~re assumed to be infinite cylinders, and graphi te gr~ins opt ical ly anisotropic spheres. On this b~sis, W]CK~AMASI~G~aE finds a degree of a l ignment of 60%, which could be as low as 30%, depending on the rat io of the ext inct ions b y silicates and by graphite.

3"2. - Expe r imen ta l measuremen t s (~7) show, on the one hand, t h a t fo rmulae for infinite cylinders m a y be ex t rapo la ted with reasonable accuracy to finite cylinders and spheroids. On the other hand, one finds nonpred ic ted s t rong resonance effects, with a large increase in the cross-section an i so t ropy up to values as large as 5 t imes those found for infinite cylinders. This would imp ly a decrease of the required degree of a l ignment b y ~ fac tor 5 ( t h a t is down to (15--10)Yo and less). Unfo r tuna t e ly exper imen ta l work on this subje t is still scarce.

3"3. - Neglect ing resonance effects, in the following we adop t ro t a t ion ellipsoidal particles, with rat ios x ~ A a / A t Of the s y m m e t r y axis and t rans- verse axis radii, and y = Ja/Jt of the corresponding m o m e n t s of inert ia. This seems to be justified also on the basis of the observed shape of ex t ra te r res t r i a l particles collected in high-al t i tude collection e• (is).

Let us now examine if the v,~lue obt~ined b y "~VICKRAMASI~,'GI[E (16) for the degree of al ignment , as high as 60%, is consis tent wi th plausible astro- physical conditions. We refer to the magne t ic mechanism. Several fo rmulae have been worked out, which give the dependence of a l ignment on the magne t ic - field s t rength, on the grain magne t ic proper t ies and on the densi ty and tem- pera ture of the gas component of in ters te l lar m a t t e r (1~).

I n the appl icat ion of those equat ions two difficulties ~rise. Firs t , t h e y give only app rox ima te results, as shown b y more refined computa t ions in which a Monte Carlo me t hod was used to s tudy the a l ignment (1,~). Second, t h e y require a knowledge of the magne t i c proper t ies of grains, which are poor ly (if a t all) known. Only an order -of -magni tude es t imate can be obtained.

(17) j . M. GREENBER(~, N. E. PEDERSEN and J. C. P]~DERSEN" Journ. Appl. Phys., 32, 231 (1961); see also ref. (el). (is) C. L. HEMENWAY, D. S. HALLGREENS and D. C. SHMULBERGER: Nature, 238, 256 (1972). For the relationship between interstellar and planetary dust see J. M. GREEN- BERG: Space Research, Vol. 9 (COSPAR) (Amsterdanl, 1969), p. 111. (19) From the vast literature we quote: a) L. DAvis and J. GR~E~ST~I~r Astrophys. Journ., 114, 206 (1951); b) R. V. JONES and L. SPITZER: Astrophys. Journ., 147, 943 (1967); c) M. PURCELI= and L. SPITZER: Astrophys. Journ., 167, 31 {1971); d) M. PUR- CELL: Physiea, 41, 100 (1969); e) P. G. MARTIN: Mont. Not. Roy. Astr. Soc., 153, 279 (1971).

~ 0 2 S. AIiELLO, A. B O N ~ T T I and F. /YIENCARAGLIA.

To our purpose it is enough to calculate the a l ignment p a r a m e t e r F for magne t ic -a l ignment processes, as g iven b y (~9~), in t roducing app rop r i a t e values for the various paramete rs

(3.3.1) F = 1 / 3 - - <cos 2 0> = (3/2q) (~ + 1 q(~ - - 1 ) ,

where 0 is the angle between magne t i c field and grain s y m m e t r y axis: q is a funct ion of the quanti t ies in brackets , g iven in (~gb); y is the ra t io of axial and t ransverse momen t s of inert ia; 5 is a p a r a m e t e r giving the re la t ive s t r eng th of the magnet ic-or ient ing mechan i sm wi th respect to the disorient ing one (gas collisions):

B~Z"(~)I~ ~- C a,~n,,(m~T~k)�89 ;

B is the magnet ic field; n, , m, and T~ the gas (essentially hydrogen) densi ty , mass of individual part icles and t e m p e r a t u r e ; Te and aa are the dust grain t e m p e r a t u r e and mean radius; Z"(m) is the magne t i c suscept iv i ty of the grains;

is the ro ta t ional f requency of the grains; C is a p a r a m e t e r depending on the grain shape (~9~); k is the B o l t z m a n n cons tan t ; 5 is of the order of uni ty .

F r o m (3.3.1) one finds the following re la t ionship be tween the a l ignment pa rame te r F and the fract ional a l ignment A of grains:

prolate grains: F ~ (1/3) A ,

oblate grains: F = - - (2/3) A .

So, for prola te grains F ranges f rom 0.2 for 60 ~ a l ignment to 0.017 for 5 ~o al ignment.

3"4.1 - F r o m synchrot ron emission of re la t ivis t ic electrons, the magne t i c - field in tens i ty B in the Galaxy has been found to be in the m e a n abou t 3 .10 -6 G. I n some dense clouds magnet ic fields as large as 10 -5 G have been measured (:o). Since it seems unlikely t ha t such large pecul iar values could be a t t r i b u t e d to ex tended regions of the Galaxy, we adop t the average value of 3-10 -6 G. The magnet ic field is t aken as helicoidal along the spiral arms, wi th the m a j o r componen t parallel to the axis of the a r m (~), ~par t f rom effects which m a y arise locally (~2), mainly, bu t not exclusively, where the densi ty of m a t t e r is large.

(20) G. L. V~nSCHUUn: Astrophys. Journ., 165, 65i (1971). (21) Reviews papers are: a) D. G. W~NTZ~L: Magnetic ]ields and galactic structure, in Ann. Rev. Astron. Astrophys., 1 (1963); b) 17. F. GARDNER and J. B. 3r The polarization of cosmic radio waves, in Ann. Rev. Astron. Astrophys., 4 (1966); e) H. C. VAN DE HULST: Observing the magnetic field, i~l Ann. Rev. Astron. Astrophys., 5 (1967). (~2) C. Cr. T. HASLAM, F. D. KAHN and J. MEA•Un•: Astron. Astrophys., 12, 388 (1971).

I N F R A - R E D POLARIZED EMISSION FROM ANISOTROPIC COSMIC DUST 3 0 3

3"4.2. - The grain t e m p e r a t u r e Te is crit ical for a l ignment . I t depends s t rongly on the chemical composit ion. We adop t here the m i x t u r e suggested b y NA~)Y and WICKRAMAS~GHE (23), which consists of iron, g raph i te and silicate particles. Grain dimensions are d is t r ibuted according to

where

](a) da z a ~ exp [-- (1/2) a/a,,,] 3 ,

a.m (graphite) ---- 0.05 ~ m ,

a~ (iron) = 0.01 ~zm,

a,, (silicate) ---- 0.15 [zm.

Grain t empera tu re s for var ious opt ical dep ths have been c o m p u t e d in (~). For the general interstel lar m e d i u m in the presence of the in ters te l lar rad ia t ion field, and for spherical particles, t empe ra tu r e s are

T ,~ph~,o = (28 • 1) ~ T , , o : (63 • 1) ~ T,,,~ : 6 ~

The quoted limits depend on the re la t ive f rac t ional contents of graphi te , iron and silicates. Pos tponing a fuller discussion to Subsect. 4"2, we r e m a r k here t h a t the values a re not ve ry different f rom those for ro t a t i on ellipsoidal particles. A typica l emission spec t rum is shown in Fig. 4 (see also Subseet. 4"4).

I ron grain t e m p e r a t u r e is p robab l y too high for subs tan t ia l a l ignment to t ake place. As for graphi te , d iamagne t ic processes are not sufficient to al ign; it has been suggested t ha t hydrogen impuri t ies could give rise to p a r a m a g n e t i c resonance effects (19~). While there is no evidence against the presence of such impuri t ies (in fact the emission fea ture a round 10 txm, usual ly a t t r i b u t e d to silicates, could be due as well to d i r ty graphi te (~4)), the higher t e m p e r a t u r e of graphi te wi th respect to silicates seems to work against the possibi l i ty t h a t graphi te contr ibutes to a large degree to polarization. I n conclusion, silicates are expected to be the main polarizing agents, as the coolest componen t of dust.

3"4.3. - For pa ramagne t i c processes Z"(w)=~(~) /3kT: this expression is independent of the par t icular grain. More refined computa t ions show t h a t this value differs f rom exact ones b y no more t h a n ( 1 0 - - 1 5 ) % (19d).

3"4.4. - T~ and n~ are those associated wi th the two-phase s t ruc tu re of gas in H I regions. Observat ions (15b) give T~ > 500 ~ and n~--0.5 cm -3 for the hot ((intercloud~) phase; T~-070 ~ (actual ly be tween 20 and 250 ~

(2a) iNT. C. WIKRAMASINGHE and K. NANDY: Nature, 227, 51 (1970). (24) N. C. WIKR~_MASI~GnE: Nature, 223, 459 (1969).

3 0 ~ 8. AIELLO, A.. B O N E T T I and E. MENCARAGLIA.

and n g ~ 1 0 em -a for t he cold c l u m p y phase. T h e o r e t i c a l models (see f.i. (15c.~))

give r e spec t ive ly T g = 8000 ~ a n d n ~ = ( 0 . 1 + 0 . 5 ) c m -3, a n d T ~ = ( 2 0 + 7 0 ) ~

a n d n ~ = (10- -100) cm -a. Tab l e I c o n t a i n s t h e va lues a d o p t e d in t he fo l lowing

ca lcu la t ions ; these va lues are l ike ly to be t h e e x t r e m e va lues expec ted for

each phase .

T A B L E I .

ng :T o ~IC, E ~ % alignement

prolate oblate prolate oblate

60 70 0.056 1.2.10 -a - - 1.5.10 -a 0.36 0.22

10 70 0.335 5.5-10 -a - - 7.5.10 -a 1.65 1.1

0.5 8000 0.626 9.0.10 -a - - 1.3.10 -2 2.7 2

0.1 8000 3.13 2.1 �9 10 -2 - - 3.7- l0 -~ 6.3 5.5

3"5. - W e t h e r dus t is m i x e d in c o m p a r a b l e a m o u n t s w i t h t he gas b o t h

in the ho t a n d in the cold phase is n o t y e t k n o w n . To our purposes i t is e n o u g h

to show how m u c h the i n t e r a c t i o n of gas wi th d u s t c o n t r i b u t e s to the a l i g n m e n t

or d i s a l i g n e m n t of grains.

F i g u r e 1 shows the p a r a m e t e r of a l i g n m e n t p l o t t e d in t e r m s of t he ra t io x

of the ax ia l a n d t r ansve r se radi i (see Subseet . 2"3), a n d of t he gas d e n s i t y

and t e m p e r a t u r e .

10 -3

10 f ( 2 / 3 ) F

10 2

/

I / I f l - - i ~ I , -

\ :

f ~ / \

/ I , & j i i , , I 10 " 10 ~ 10 ~ x

Fig. 1. - Degree of alignment for dust grains as a function of the ratio between axial and transverse axis length. Solid lines denote the hot, thin regions; long-dashed lines denote cold, dense regions. Also shown is (+ ) the degree of al ignment as found by WICKRAMASINGHE (see text).

IN:PRA-RED :POLARIZED ~MISSION :PROM ANISOTROPIC COSMIC DUST 3 0 5

The results are summar ized in Table I , where the m ~ x i m u m vglue of the f ract ional a l ignment is repor ted in correspondence of the adop ted choice of n~ and T~.

One obtains values definitely smaller t h a n t h a t found b y WICKI~A~CIASI:NGItE f rom the observed polar izat ion (16).

Those low values can be compared with the lower l imits suggested in Subseet. 3"2. I n fact , in the present computa t ions we did not t ake into account possible resonance effects due to the shape (~7), or peculiar magne t i c prop- erties (,gb), of the grains.

F r o m Table 1 and Fig. 1 one notices:

a) Al ignment is f avoured in hot , th in regions (however, in cold dense regions the dependence on the magne t i c field m a y be stronger).

b) Prola te grains are favoured compared with oblate grains. Fu r t e rmore , the degree of ~l ignment reaches a m a x i m u m for x----3 or x ~ 0.4 respect ively, and decreases for very elongated or f la t tened shapes.

4. - Infra-red polarized emiss ion f rom dust grains.

4 " 1 . - We computed the emission cross-sections of ro t a t ion ellipsoidal

grains in the far infra-red using the a p p r o x i m a t e relat ions (2~)

2 ~ V el ~) (4.1.1) Sj = i - (4J)(P~/4=) + 1)~ + (4~)(P~/4=)) ~ ' J = ~' t ,

where V is the grain volume, P is the depolarizing factor , e = 61 - - i e . 2 is the complex dielectric constant and indices a, t refer to the electric vec tor paral lel or normal to the s y m m e t r y axis. The ra t io Sa/St is shown in Fig. 2 for dif- ferent values of the optical constants .

4"2. - The to ta l emission f rom anisotropic grains will be in general dif- ferent f rom tha t f rom spherical grains. This leads, for a given rad ia t ion field, to different grain t empera tures . I n par t icu lar cases it has been shown t h a t the t e m p e r a t u r e of isotropic ellipsoidal gTains will be sl ightly lower t h a n t h a t of spherical grains (26).

Similar considerations app ly to opt ical ly anisotropic dust grains. F o r instance, let us consider graphi te grains for which the conduc t iv i ty along the c-axis is abou t 150 t imes less t h a n along the basal plane. I f we assume for s implici ty spherical grains at a t e m p e r a t u r e a round 30 ~ the emission will

(2~) R. GANS: Ann. der Phys., 47, 270 (1915). (26) j . M. CrREENBERG and G. A. SHAH: Astron. Astrophys., 12, 250 (1971).

~ ) ~ S . A I E L L O ~ A , B O N E T T I a n d F . M E N C A R A G L I &

I I i- t

1 0 1 ~

t,3 ~

,-~ 10 o

101

I

I

J ++ j / ' , J / J

/ /

/ f �9 " J

/ v

i / j 1 0 ~ 10 ~ x 1 O!

Fig. 2. - Ratio between axial and transverse cross-sections as a function of the ratio between axial and transverse axis length for different optical constants ,h - - 1 . 5 - - i k ; long-dashed line k = 0.1, dot-dashed line k--0 .5 , short-dashed line k = 1, solid line k = 5.

Cake place at w a v e l e n g t h s greater t h a n 80 ~m; at t h e s e w a v e l e n g t h s equa-

t i o n (4.1.1) becomes , s ince e<~)= 2(~j)L/c, v 2

t h a t is

S j - - 2 ~ V 2( l j2 /e _ A 2 - 2 2 (2a~2 /c ) 2 a~ '

St ~ 50 So

(at is the e lectric c o n d u c t i v i t y ) .

INFRA-RED POLARIZED EMISSION FROM A.NISOTROPIC COSMIC DUST 3D7

Graphi te t empera tu re , which has been compu ted in Subsect. 3"~.2 assuming an average cross-section S ~ S o, should be corrected, t ak ing into account tha~

s - - (2so + s,)/3 = ~7 so .

This large increase in emission efficiency leads to a decrease in t e m p e r a t u r e of abou t 30 %.

Similar considerations could be appl ied to silicate grains, bu t in this case it can shown tha t , within the l imits of our approx imat ions , t he opt ical aniso- t r o p y is less impor t an t t h a n the shape anisotropy.

So, in the case of bo th geometr ical and opt ical anisot ropy, t he emissio~ spec t rum will be shifted towards grea te r wavelengths . F u r t h e r m o r e , since the t e m p e r a t u r e will depend on the specific shape, and it is to be expec ted t h a t there will be a dis t r ibut ion of ellipticities, the ac tua l spec t rum will be a super- posit ion of spect ra corresponding to different t empera tu re s , and as a conse- quence it will have an enlarged width. These fea tures of the emission spectra l d is t r ibut ion m a y be difficult to observe and in te rpre t (see f.i. (27)).

4"3. - STEIN first pointed out (1~) t h a t t he rma l emission f rom anisotropic dus t grains is polarized. For a single grain the m a x i m u m degree of polar- izat ion P ' is given by

P'~- ( S ~ - S,)I(Sa-~- St).

This result alone cannot be used to obta in an es t ima te of the degree of polar izat ion f rom n cloud of dust grains, because one has to t ake into account the degree of al ignment.

Denot ing again with indices j ~ a, t the directions a long and normal to the magnet ic field, we have for the to ta l emi t t ed power

I j ~ 1 - - cxp [-- ~r ,

where Tj is the optical dep th for radia t ion polar ized along the j-direction. Expressions for vj are given in (19a). I t is found t h a t t he f rac t ional polar-

izat ion p f rom a dust cloud of a l ignment p a r a m e t e r F is g iven b y

3 ( S o - - S~) F cos ~ P = ~ {(2s. + ,~)/3} + ( s , - - s~ ) (F /2 ) (a - -3 e o s ~ ) '

where v + 90 ~ is the angle be tween the magne t i c field and the direct ion of observat ion.

(27) E. F. EtCICKSON, C. D. SWIFT, F. C. WITTEBORN, A. J. MORD, G. C. AUGASON, L. J. CAROFF, L. W. KUNZ and L. P. GIVER: Astrophys. Journ., 183, 535 (1973); D. Y. GEZARI, R. R. JOYCE, G. RIGIIINI and M. SIMON: preprint, to be published in Astro- phys. Jouvn.

~ 0 ~ S. AI]~LLO, A. B O N E T T I and F. MENCAICAGLIA.

Figure 3 shows p vs. S,~/St for v ~ 0% V~Te m a y outline some interest ing conclusions. For both oblate and prolate optically isotropic spheroids p < 0, tha t is, the direction of polarization (*) is normal to the magnet ic field:

oblate spheroids : /~ < 0

prolate spheroids : /~ > 0

Sa-- S t y 0 , p ~ 0 ,

S ~ - - S t < O , p < 0 .

P

10

10

o ~

a)

b)

a)

e)

~)

CL:

b)

c)

I

10 -2 10 -1 10 0 101 x 10 2

Fig'. 3. - Degree of polarization of radiation emitted by a cloud of dust grains oblate (o) or prolate (p), as a function of the ratio between axial and transverse axis length; values are shown for different degrees of alignement: a) 100%, b) 50%, c) 25~ d) 12.5%, e) 6.2590 , /) 3.125~ .

This means than the emitted radiat ion is polarized normal to the polarized component of starlight, and parallel to the synchro t ron polarizat ion plane.

This is the case for prolate silicate grains. Optically anisotropic graphite flakes ~re oriented as oblate spheroids (F < 0),

but (see Subsect. 4"2) S a - - S t < 0, so tha t p > 0; the direction of polarizat ion of the emit ted radiation is p~rallel to the direction of magnet ic field and

starlight polarization, tha t is normal to t ha t of the radiat ion emit ted by silicates. As a consequence, when making broad-band polar imetry, the me~surable par t

(*) To avoid confusion we adopt the convention that theplane of polarization is the plane containing the direction of propagation and the direction of the electric vector (D. CLARK]~ and J. F. GRAINGER: Polarized Light and Optical Measurement (London, 1971), p. 173).

INFRA-RED FOLARIZED EMISSION FROM ANISOTROPIC COSMIC DUST 309

of the polarized radiat ion decreases. Apar t the over-simplifications which are introduced in the computa t ion of the alignment parameter , we m a y note t ha t F (and so the polarization degree p) depends on the gas densi ty and tem-

perature and on the magnetic field within the part icular cloud or set of clouds which are along the line of sight. Changes in these parameters give rise to differences in the resulting p. ~( Local ~) variations of T~ and n, and of B are

a possible reason for the larger scat ter of starlight polarization than of extinc- tion measurements.

4 " 4 . - Following the model of dust distr ibution adopted herewith (see Subsect. 2"2), the far II~ flux should consist a) of a background emission f rom the quasi-uniform low-density large-scale component and b) of the emission

from (( local )) sources (dark, or more generally dusty~ nebulae of various kinds).

4"4.1. - For the large-scale, dilute component , assuming a composit ion as in Subsect.s 3"4.2 and 4"2, and an average grain density of 10 -14 cm -~, con- sistent with the average extinction of 1 magni tude per kpc, one obtains a

background emission between 30 and 500 ~tm (*) of (10 ~~ W cm -2 sr -*, with interstellar light as the only heat ing source.

This result is not in contradict ion with the measured background flux (1 - - 2 ) 1 0 -1~ W era. -2 s r -1 (2s).

]f we assume for silicates complete al ignment tdong a direction perpendicular to the line of sight, and for graphite an alignment of I r so tha t p . ~ 50 o/o

and p,,~h--~1%, the upper limit of the expected polarized flux is around (10-14--10 -~5) ~ , with a geometric factor of 10 - : e m ~ sr.

4"4.2. - For a dense, quasi-spherical, structureless cloud (extinction 8 mag- nitudes, radius 7 pcs, distance 200 pcs) the integrated flux at Ea r th is shown

in Table I I . wl, w2, w3 are the relative weights of the optical depths of graphite,

TABLE I I .

~1 % w~ w1 w~ w~ wtot = w1 + w~ + w .

a) 1 0.66 1 6.4 3.7 0.6 10.7

b) 3 1.66 1 12.9 4.0 0.5 17.4

c) 7 2.66 1 17.4 5.7 0.2 23.3

iron, silicate at the wavelength of 0.45 ~zm (~3), and W1, W2, W3 are the in tegra ted fluxes from 30 to 500 ~zm in units of :10 -13 W cm -~. The silicate grain densi ty has been assumed to be 10 -11 cm-~; again no other heat ing sources are present

(*) Beyond this limit the universal black-body radiation becomes important. (2s) HARWIT et al., from K. SHIVANANDAN: private communication.

310 8 . . & I ] ] L L O ~ A. BON]~TTI and F. MENCARAGLIA.

except the general interstellar radiat ion field. A typical emission spec t rum

is shown in Fig. 4, which refers very nearly to ease b) of Table I I (see also

Subsect. 3"4.2).

I i

16 I- L

10 ]

\\ - - ' ~ y-

] ',

/ ] ',

/

i / 1

I I i ~ L L _ _ _ _

10 2

\/ / \

/ \ ,1 /, \

, \

pm 10 3

Fig. 4. - Spectral emission of a cloud composed of graphite, iron and silicates grains with temperatures of 25 ~ 60 ~ 7 ~ T~ = 60 ~ Tg = 25 ~

Ts~_7 ~

If we assume again for silicates complete al ignment along a direction perpen-

dicular to the line of sight, and for graphi te an alignment of 1 ~o (P~il~ 50 %, P~,~h~ 1 ~ a value of 10 -15 W cm -~ m a y be retained as an upper limit for

the polarized flux from the adopted dark nebula. I n conclusion~ polarimetric measurements of dus ty regions in the far infra-

red appear to be within the limits or present detector performances, provided

tha t the spurious polarization of the measuring device is kept as low as 1%.

4"5. - I n the evaluation of the amoun t of the emit ted polarized radiat ion,

allowance should be made also for the polarization in t roduced by the grains located between the emitt ing region and the observer. Assuming again for

INFRA-RED POLARIZED EMISSION FROM ANISOTROPIC COSSIIC DUST 311

simplici ty t h a t silicates are comple te ly aligned and t h a t the emission f r o m the other grains is unpolarized, we obta in for the degree of polar iza t ion acqu i red by the radia t ion t ravel l ing th rough the geometr ica l p a t h L within the cloud

p~= (So- S~)N,,L,

where the meaning of symbols is obvious. This polar iza t ion is paral lel to s tar - light polarization. Assuming in the spec t ra l region of in teres t Sa.t~ 5-10 -1~ cm 2 (efficiency fac tor abou t 10-~), we have for the same cloud as before an u p p e r

l imit

P Z = 5 " 1 0 - 1 2 e r a 2 " 1 0 - 1 1 c m - 3 " 4 X 3 - 1 0 is cm = 6 " 1 0 - 4 .

So, the polarizing effect of aligned silicates on the rad ia t ion emi t t ed b y grains is negligible compared with the a m o u n t of d i rec t ly emi t t ed polar ized radiat ion. I t should be kep t in mind t h a t a f rac t ion of the emi t t ed r ad ia t ion through a polarizing birefr ingent cloud should be circular ly polarized (29).

A final r e m a r k is tha t , because of the t r a n s p a r e n c y of in ters te l lar m a t t e r in the I R , it is to be expected to have in the field of view of the measu r ing device a grea ter n u m b e r of sources t h a n in the case of visible light. This will cause a decrease of the degree of polar izat ion of the observed radia t ion, ana logue to the decrease of p/Av vs. distance in the case of visible l ight (A~ ex t inc t ion

in the visible) (1~).

5 . - C o n c l u s i o n s .

5"1. - An observable a m o u n t of Ii~ rad ia t ion f r o m aligned dust grains is expected to be polarized. So, I R polar imetr ic m e a s u r e m e n t s can be useful in the s tudy of the opaque componen t of inters te l lar ma t t e r . T h e y con t r ibu te to the knowledge of the na ture and dis t r ibut ion of the different kinds of dus t grains b y ident i fying the corresponding b road ly different spect ra l regions of emission. Also t hey give more precise es t imates of the degree of a l igmnent , as computed cross-sections are more accura te in the long-wavelength region. Observat ions can be extended to the regions poor of s tars (such as h igh- la t i tude regions or heavi ly obscured fields), as an extension of visible po la r imet r ic measurements . I n some cases t he rm a l and synchro t ron rad ia t ion could be distinguished b y ident ifying the plane of polar iza t ion wi th respect to t h a t of s tar l ight wi thout resort ing to the spectra l measurements .

(29) p. G. MARTIn: Mont..Not. Roy. Astr. Soc., 159, 179 (1972).

312 S. AIELLO, A. BONETTI and F. M]~NCARAGLI&

5 " 2 . - F u r t h e r m o r e ~nfra- red p o l a r i m e t r y m a y a d d i n f o r m a t i o n on t h e

l a rge - sca l e d i s t r i b u t i o n of t h e g a l a c t i c m a g n e t i c f ield, as l ong as t h e m a g n e t i c

m e c h a n i s m can be cons ide r ed eff ic ient e n o u g h as to e x p l a i n t h e l a r g e - s c a l e

r e g u l a r i t i e s of s t a r l i g h t p o l a r i z a t i o n . A n a d v a n t a g e of I R o v e r r a d i o p o l a r i m e t r i e

m e a s u r e m e n t s cou ld be t h e neg l ig ib le i m p o r t a n c e of t h e F a r a d a y effect . I n d e e d

t h e ang le of r o t a t i o n of t h e p o l a r i z a t i o n p l a n e is p r o p o r t i o n a l to t h e i n v e r s e

s q u a r e of t h e r a d i a t i o n f r e q u e n c y . So II~ m e a s u r e m e n t s m a y b e u se fu l in

s t u d y i n g reg ions w i t h i n or b e y o n d s t r o n g m a g n e t i c f ields.

5"3. - F r o m a t e c h n i c a l p o i n t of v iew, i t m a y b e r e m e m b e r e d t h a t p o l a r -

i m e t r i c (II~) m e a s u r e m e n t s do n o t r e q u i r e ~ c o m p a r i s o n w i t h b a c k g r o u n d ,

s ince t h e b e a m m o d u l a t i o n is o b t a i n e d b y c o m p a r i n g f luxes in t w o d i f f e r en t

p o l a r i z a t i o n s t a t e s .

�9 R I A S S U N T O

Si ~ analizzato il p rob lems degli effett i polar imet r ic i della polvere cosmica, con par t icolare r iferimento al meccanismo magnetico di al l ineamento. Grani anisotropi al l ineati emettono nell ' infrarosso radiazione parz ia lmente polar izza ta nei l imi t i di sensibilit/~ degli a t tua l i r ivelatori . Misure polar imetr iche nell ' infrarosso possono con- t r ibui re ut i lmente alla conoscenza de t t ag l i a ta della na tu ra e della dis tr ibuzione della polvere cosmica.

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